Báo cáo hóa học: " Research Article Channel Equalization in Filter Bank Based Multicarrier Modulation for Wireless " doc

In a companion paper [25], a similar subband equalizer structure is applied to the filter bank approach for frequency domain equalization in single carrier transmission.. The chosen filt

Volume 2007, Article ID 49389, 18 pages

doi:10.1155/2007/49389

Research Article

Channel Equalization in Filter Bank Based Multicarrier

Modulation for Wireless Communications

Tero Ihalainen, 1 Tobias Hidalgo Stitz, 1 Mika Rinne, 2 and Markku Renfors 1

1 Institute of Communications Engineering, Tampere University of Technology, P.O Box 553, Tampere FI-33101, Finland

2 Nokia Research Center, P.O Box 407, Helsinki FI-00045, Finland

Received 5 January 2006; Revised 6 August 2006; Accepted 13 August 2006

Recommended by See-May Phoong

Channel equalization in filter bank based multicarrier (FBMC) modulation is addressed We utilize an efficient oversampled filter bank concept with 2x-oversampled subcarrier signals that can be equalized independently of each other Due to Nyquist pulse shaping, consecutive symbol waveforms overlap in time, which calls for special means for equalization Two alternative linear low-complexity subcarrier equalizer structures are developed together with straightforward channel estimation-based methods to calculate the equalizer coefficients using pointwise equalization within each subband (in a frequency-sampled manner) A novel structure, consisting of a linear-phase FIR amplitude equalizer and an allpass filter as phase equalizer, is found to provide enhanced robustness to timing estimation errors This allows the receiver to be operated without time synchronization before the filter bank The coded error-rate performance of FBMC with the studied equalization scheme is compared to a cyclic prefix OFDM reference

in wireless mobile channel conditions, taking into account issues like spectral regrowth with practical nonlinear transmitters and sensitivity to frequency offsets It is further emphasized that FBMC provides flexible means for high-quality frequency selective filtering in the receiver to suppress strong interfering spectral components within or close to the used frequency band

Copyright © 2007 Tero Ihalainen et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

1 INTRODUCTION

Orthogonal frequency division multiplexing (OFDM) [1]

has become a widely accepted technique for the realization

of broadband air-interfaces in high data rate wireless

ac-cess systems Indeed, due to its inherent robustness to

multi-path propagation, OFDM has become the modulation choice

for both wireless local area network (WLAN) and terrestrial

digital broadcasting (digital audio and video broadcasting;

DAB, DVB) standards Furthermore, multicarrier

transmis-sion schemes are generally considered candidates for the

fu-ture “beyond 3 G” mobile communications

All these current multicarrier systems are based on the

conventional cyclic prefix OFDM modulation scheme In

such systems, very simple equalization (one complex

coef-ficient per subcarrier) is made possible by converting the

broadband frequency selective channel into a set of

paral-lel flat-fading subchannels This is achieved using the inverse

fast Fourier transform (IFFT) processing and by inserting a

time domain guard interval, in the form of a cyclic prefix

(CP), to the OFDM symbols at the transmitter By

dimen-sioning the CP longer than the maximum delay spread of the

radio channel, interference from the previous OFDM sym-bol, referred to as inter-symbol-interference (ISI), will only affect the guard interval At the receiver, the guard interval

is discarded to elegantly avoid ISI prior to transforming the signal back to frequency domain using the fast Fourier trans-form (FFT)

While enabling a very efficient and simple way to com-bat multipath effects, the CP is pure redundancy, which de-creases the spectral efficiency As a consequence, there has recently been a growing interest towards alternative multi-carrier schemes, which could provide the same robustness without requiring a CP, that is, offering improved spectral efficiency Pulse shaping in multicarrier transmission dates back to the early work of Chang [2] and Saltzberg [3] in the sixties Since then, various multicarrier concepts based

on the Nyquist pulse shaping idea with overlapping sym-bols and bandlimited subcarrier signals have been developed

by Hirosaki [4], Le Floch et al [5], Sandberg and Tzannes [6], Vahlin and Holte [7], Wiegand and Fliege [8], Nedic [9], Vandendorpe et al [10], Van Acker et al [11], Siohan

et al [12], Wyglinski et al [13], Farhang-Boroujeny [14,15], Phoong et al [16], and others One central ingredient in the

later developments is the theory of efficiently implementable,

modulation-based uniform filter banks, developed by

Vet-terli [17], Malvar [18], Vaidyanathan [19], and Karp and

Fliege [20], among others In this context, the filter banks are

used in a transmultiplexer (TMUX) configuration

We refer to the general concept as filter bank based

multi-carrier (FBMC) modulation In FBMC, the submulti-carrier signals

cannot be assumed flat-fading unless the number of

subcar-riers is very high One approach to deal with the fading

fre-quency selective channel is to use waveforms that are well

lo-calized, that is, the pulse energy both in time and frequency

domains is well contained to limit the effect on consecutive

symbols and neighboring subchannels [5,7,12] In this

con-text, a basic subcarrier equalizer structure of a single complex

coefficient per subcarrier is usually considered Another

ap-proach uses finite impulse response (FIR) filters as subcarrier

equalizers with cross-connections between the adjacent

sub-channels to cancel the inter-carrier-interference (ICI) [6,10]

A third line of studies applies a receiver filter bank structure

providing oversampled subcarrier signals and performs

per-subcarrier equalization using FIR filters [4,8,9,11,13] The

main idea here is that equalization of the oversampled

sub-carrier signals restores the orthogonality of the subsub-carrier

waveforms and there is no need for cross-connections

be-tween the subcarriers This paper contributes to this line of

studies by developing low-complexity linear per-subcarrier

channel equalizer structures for FBMC The earlier

contri-butions either lack connection to the theory of efficient

mul-tirate filter banks, use just a complex multiplier as subcarrier

equalizer or, in case of non trivial subcarrier equalizers, lack

the analysis of needed equalizer length in practical wireless

communication applications (many of such studies have

fo-cused purely on wireline transmission) Also various

practi-cal issues like peak-to-average power ratio and effects of

tim-ing and frequency offsets have not properly been addressed

in this context before

The basic model of the studied adaptive sine

modu-lated/cosine modulated filter bank equalizer for

transmul-tiplexers (ASCET) has been presented in our earlier work

[21–23] This paper extends the low-complexity equalizer

of [23,24], presenting comprehensive performance analysis,

and studies the tradeoffs between equalizer complexity and

number of subcarriers required to achieve close-to-ideal

per-formance in a practical broadband wireless communication

environment A simple channel estimation-based calculation

of the equalizer coefficients is presented The performance of

the studied equalizer structures is compared to OFDM,

tak-ing into account various practical issues

In a companion paper [25], a similar subband equalizer

structure is applied to the filter bank approach for frequency

domain equalization in single carrier transmission In that

context, filter banks are used in the analysis-synthesis

config-uration to replace the traditional FFT-IFFT transform-pair

in the receiver

The rest of the paper is organized as follows.Section 2

briefly describes an efficient implementation structure for

the TMUX based on exponentially modulated filter banks

(EMFB) [26] The structure consists of a critically sampled

synthesis and a 2x-oversampled analysis bank The problem

of channel equalization is addressed inSection 3 The theo-retical background and principles of the proposed compen-sation method are presented The chosen filter bank struc-ture leads to a relatively simple signal model that results in criteria for perfect subcarrier equalization and formulas for FBMC performance analysis in case of practical equalizers

A complex FIR filter-based subcarrier equalizer (CFIR-SCE) and the so-called amplitude-phase (AP-SCE) equalizer are presented Especially, some low-complexity cases are ana-lyzed and compared in Section 4 In Section 5, we present

a semianalytical and a full time domain simulation setup

to evaluate the performance of the equalizer structures in a broadband wireless communication channel Furthermore, the effects of timing and frequency offsets, nonlinearity of

a power amplifier, and overall system complexity are briefly investigated Finally, the conclusions are drawn inSection 6

2 EXPONENTIALLY MODULATED PERFECT RECONSTRUCTION TRANSMULTIPLEXER

Figure 1 shows the structure of the complex exponen-tially modulated TMUX that can produce a complex in-phase/quadrature (I/Q) baseband signal required for spec-trally efficient radio communications [23] It has real format for the low-rate input signals and complex I/Q-presentation for the high-rate channel signal It should be noted that FBMC with (real) m-PAM as subcarrier modulation and

OFDM with (complex)m2-QAM ideally provide the same bit rate since in general the subcarrier symbol rate in FBMC

is twice that of OFDM for a fixed subchannel spacing In this structure, there are 2M low-rate subchannels equally spaced

between [− F s /2, F s /2], F sdenoting the high sampling rate EMFBs belong to a class of filter banks in which the subfilters are formed by frequency shifting the lowpass pro-totypeh p[n] with an exponential sequence [27] Exponen-tial modulation translates H p(e jω) (lowpass frequency re-sponse) around the new center frequency determined by the subcarrier index k The prototype h p[n] can be optimized

in such a manner that the filter bank satisfies the perfect-reconstruction (PR) condition, that is, the output signal is

a delayed version of the input signal [27, 28] In the gen-eral form, the synthesis and analysis filters of EMFBs can be written as

f k[n] =



2

M h p[n] exp



j



n + M + 1

2



k +1

2



π M



, (1)

h k[n] =



2

M h p[n] exp



− j



N − n + M + 1

2



k +1

2



π M



, (2) respectively, wheren =0, 1, , N and k =0, 1, , 2M −1 Furthermore, it is assumed that the filter order isN =2KM −

1 The overlapping factorK can be used as a design

parame-ter because it affects on how much stopband attenuation can

be achieved Another essential design parameter is the stop-band edge of the prototype filterω = (1 +ρ)π/2M, where

CMFB synthesis

SMFB synthesis

x k[m]

x2M 1 k[m]

F M(ω)

π

F2M 1(ω)F0 (ω) F1 (ω) F M 1(ω)

Channel

Re 

Im 

CMFB analysis

SMFB analysis CMFB analysis

SMFB analysis

+

1/2

+

1/2

j

+

+

+ +

+ I Q

I Q

SCE

SCE

Re 

Re 



x k[m]



x2M 1 k[m]

Figure 1: Complex TMUX with oversampled analysis bank and per-subcarrier equalizers

the roll-off parameter ρ determines how much adjacent

sub-channels overlap Typically, ρ = 1.0 is used, in which case

only the contiguous subchannels are overlapping with each

other, and the overall subchannel bandwidth is twice the

sub-channel spacing

In the approach selected here, the EMFB is implemented

using cosine and sine modulated filter bank (CMFB/SMFB)

blocks [28], as can be seen inFigure 1 The extended lapped

transform (ELT) is an efficient method for implementing PR

CMFBs [18] and SMFBs [28] The relations between the

syn-thesis and analysis filters of the 2M-channel EMFB and the

correspondingM-channel CMFB and SMFB with the same

real FIR prototypeh p[n] are

f k[n] =

⎧

⎨

⎩

f k c[n] + j f k s[n], k ∈[0,M −1]

−f2Mc −1− k[n] − j f2Ms −1− k[n]

, k ∈[M, 2M −1],

(3)

h k[n] =

⎧

⎨

⎩

h c

k[n] − jh s

−h c

2M−1− k[n]+ jh s

2M−1− k[n]

, k ∈[M, 2M −1],

(4) respectively A specific feature of the structure inFigure 1is

that while the synthesis filter bank is critically sampled, the

subchannel output signals of the analysis bank are

oversam-pled [26] by a factor of two This is achieved by using the

symbol-rate complex (I/Q) subchannel signals, instead of the

real ones that are sufficient for detection after the channel

equalizer, or in case of a distortion-free channel

We consider here the use of EMFBs which have odd

chan-nel stacking, that is, the center-most pair of subchanchan-nels is

symmetrically located around the zero frequency at the

base-band We could equally well use a modified EMFB

struc-ture [26] with even stacking (the center-most subchannel

lo-cated symmetrically about zero) The latter form has also a

slightly more efficient implementation structure, based on

DFT-processing The proposed equalizer structure can also

be applied with modified DFT (MDFT) filter banks [20],

with modified subchannel processing However, for the

fol-lowing analysis EMFB was selected since it results in the most

straightforward system model

Further, although the discussion here is based on the use

of PR filter banks, also nearly perfect-reconstruction (NPR) designs could be utilized In the critically sampled case, the implementation benefits of NPR designs are limited because the efficient ELT structures cannot be utilized [29] However,

in the 2x-oversampled case, having two parallel CMFB and SMFB blocks, the implementation benefits of NPR designs could be more significant

3 CHANNEL EQUALIZATION

The problem of channel equalization in the FBMC context

is not so well understood as in the DFT-based systems Our equalizer concept can be applied to both real and complex modulated baseband signal formats; here we focus on the complex case In its simplest form, the subcarrier equalizer structure consists only of a single complex coefficient that adjusts the amplitude and phase responses of each subchan-nel in the receiver [22] Higher-order SCEs are able to equal-ize each subchannel better if the channel frequency response

is not flat within the subchannel As a result, the use of higher-order SCEs enables to increase the relative subchan-nel bandwidth because the subchansubchan-nel responses are allowed

to take mildly frequency selective shapes As a consequence, the number of subchannels to cover a given signal band-width by FBMC can be reduced In general, higher-order equalizer structures provide flexibility and scalability to sys-tem design because they offer a tradeoff between the num-ber of required subchannels and complexity of the subcarrier equalizers

The oversampled receiver is essential for the proposed equalizer structure In case of roll-off ρ=1.0 or lower,

non-aliased versions of the subchannel signals are obtained in the 2x-oversampled receiver when complex (I/Q) signals are sampled at the symbol rate Consequently, complete chan-nel equalization in an optimal manner is possible As a result

of the high stopband attenuation of the subchannel filters, there is practically no aliasing of the subchannel signals in the receiver bank Thus perfect equalization of the distort-ing channel within the subchannel passband and transition band regions would completely restore the orthogonality of the subchannel signals [9]

3.1 Theoretical background and principles

Figure 2(a) shows a subchannel model of the complex

TMUX with per-subcarrier equalizer A more detailed model

that includes the interference from the contiguous

subchan-nels is shown inFigure 2(b) Limiting the sources of

inter-ference to the closest neighboring subchannels is justified if

the filter bank design provides sufficiently high stopband

at-tenuation Furthermore, in this model the order of

down-sampling and equalization is interchanged based on the

mul-tirate identities [19] The latter model is used as a basis for

the cross-talk analysis that follows It is also convenient for

semianalytical performance evaluations The equalizer

con-cept is based on the property that with ideal sampling and

equalization, the desired subchannel signal, carried by the

real part of the complex subchannel output, is orthogonal

to the contiguous subchannel signal components occupying

the imaginary part The orthogonality between the

subchan-nels is introduced when the linear-phase lowpass prototype

h p[n] is exponentially frequency shifted as a bandpass filter,

with 90-degree phase-shift between the carriers of the

con-tiguous subchannels

In practice, the nonideal channel causes amplitude and

phase distortion The latter results in rotation between the

I-and Q-components of the neighboring subchannel signals

causing ICI or cross-talk between the subchannels ISI, on

the other hand, is mainly caused by the amplitude distortion

The following set of equations provides proofs for these

state-ments We derive them for an arbitrary subchannelk on the

positive side of the baseband spectrum and the results can

easily be extended for the subchannels on the negative side

using (3) and (4) In the following analysis we use a

non-causal zero-phase system model, which is obtained by using,

instead of (2), analysis filters of the form

h k[n] =



2

M h p[n + N] exp



− j



− n + M +1

2



k+1

2



π M



.

(5)

By referring to the equivalent form, shown inFigure 2(b),

and adopting the notation from there, we can express the

cas-cade of the synthesis and analysis filters of the desired

sub-channelk as

f k[n] ∗ h k[n] =

b

l = a

h c k[l] f k c[n − l] +

b

l = a

h s k[l] f k s[n − l]

+j ·

 b

l = a

h c k[l] f k s[n − l] −

b

l = a

h s k[l] f k c[n − l]



= t I

k[n] + j · t k Q[n] = t k[n],

(6)

where∗denotes the convolution operation, summation

in-dexes are a = − N + max(n, 0) and b = min(n, 0), and

n ∈[− N, , N].

3.1.1 ICI analysis

For the potential ICI terms from the contiguous subchannels

k −1 andk + 1 (below and above) to the subchannel k of

interest, we can write

f k −1[n] ∗ h k[n]

=

b

l = a

h c k[l] f k c −1[n − l] +

b

l = a

h s k[l] f k s −1[n − l]

+j ·

 b

l = a

h c k[l] f s

k −1[n − l] −

b

l = a

h s k[l] f c

k −1[n − l]



= v I

k[n] + j · v Q k[n] = v k[n],

f k+1[n] ∗ h k[n]

=

b

l = a

h c k[l] f c k+1[n − l] +

b

l = a

h s k[l] f s k+1[n − l]

+j ·

 b

l = a

h c k[l] f k+1 s [n − l] −

b

l = a

h s k[l] f k+1 c [n − l]



= u I k[n] + j · u Q k[n] = u k[n],

(7) respectively

Due to PR design, the real partsv I

k[m] and u I

k[m] (m

be-ing the sample index at the low rate) of the downsampled subchannel signals are all-zero sequences (or close to zero sequences in the NPR case) So ideally, when the real part

of the signal is taken in the receiver, no crosstalk from the neighboring subchannels is present in the signal used for de-tection Channel distortion, however, causes phase rotation between the I- and Q-components breaking the orthogonal-ity between the subcarriers Channel equalization is required

to recover the orthogonality of the subcarriers

The ICI components from other subcarriers located fur-ther apart from the subchannel of interest are considered negligible This is a reasonable assumption because the ex-tent of overlapping of subchannel spectra and the level of stopband attenuation can easily be controlled in FBMC In fact, they are used as optimization criteria in filter bank de-sign, as discussed in the previous section

The cascade of the distorting channel with instantaneous impulse response (in the baseband model) hch[n] and the

upsampled version of the per-subcarrier equalizerc k[n] (see

Figure 2) applied to the subchannel k of interest can be

expressed as

hch[n] ∗ c k[n] = r k[n]. (8)

In the analysis, a noncausal high-rate impulse responsec k[n]

is used for the equalizer, although in practice the low-rate causal formc k[m] is applied.

Next we analyze the ICI components potentially remain-ing in the real parts of the subchannel signals that are used for detection.Figure 3visualizes the two ICI bands for subchan-nelk =0 We start from the lower-side ICI term and use an equivalent baseband model, where the potential ICI energy

Distorting channel

M f k[ n]

X k

h ch[ n] h k[ n] M c k[m] Re 



X k

Synthesis bank Analysis bank Equalizer

(a)

M

M

M

u I

k[n] + ju Q k[n]

= f k+1[n] h k[n]

t I

k[n] + jt Q k[n]

= f k[ n] h k[ n]

v I

k[n] + jv k Q[n]

= f k 1[n] h k[n]

X k+1

X k

X k 1

+ r I k[n] + jr k Q[n]

= h ch[n] c k[n] Re M



X k

c k[n] =

⎧

⎩c k

[n/m], forn = mM, m Z

0, otherwise (b)

Figure 2: Complex TMUX with per-subcarrier equalizer (a) System model for subchannelk (b) Equivalent form including also contiguous

subchannels for crosstalk analysis

Desired subchannel

π

2M

3π

2M

RX filter of the desired subchannel

TX filter of the contiguous subchannel

Potential ICI spectrum

Figure 3: Potential ICI spectrum for subchannelk =0

is symmetrically located about zero frequency We can write

the baseband cross-talk impulse response from subchannel

k −1 to subchannelk in case of an ideal channel as



v k[n] =  v I

k[n] + j vk Q[n] = v k[n]e − jnkπ/M (9)

In the appendix, it is shown that this impulse response is

purely imaginary, that is,v I

k[n] ≡ 0 andvk[n] = v0[n] In

case of nonideal channel with channel equalization, the

base-band cross-talk impulse response can now be written as



g k −1

wherer k[n] = r k[n]e − jnkπ/M Here the upper index denotes

the source of ICI Now we can see that if the equalized

chan-nel impulse response is real in the baseband model, then the

cross-talk impulse response is purely imaginary, and there is

no lower-side ICI in the real part of the subchannel signal

that is used for detection

At this point we have to notice that the lower-side ICI

energy is zero-centered after decimation only for the

even-indexed subchannels, and for the odd subchannels the above

model is not valid as such However, we can establish a sim-ple relation between the actual decimated subchannel output sequencez k[mM] in the filter bank system and the sequence

obtained by decimating in the baseband model It is straight-forward to see that the following relation holds:

z k[n]e − jnkπ/M

Thus, for odd subchannels, the actual decimated ICI se-quence is obtained by lowpass-to-highpass transformation (i.e., through multiplication by an alternating±1-sequence) from the ICI sequence of the baseband model Then the ac-tual ICI is guaranteed to be zero if it is zero in the baseband model Therefore, a sufficient condition for zero lower-side ICI in all subchannels is that the equalized baseband channel impulse response is purely real

For the upper-side ICI, we can first write the baseband model as



u k[n] =  u I k[n] + j uQ k[n] = u k[n]e − jn(k+1)π/M (12)

Again, it is shown in the appendix that this baseband im-pulse response is purely imaginary, that is, uI k[n] ≡ 0 and



u k[n] = u2M−1[n] With equalized nonideal channel, the

cross-talk response is now



g k k+1[n] = ju Q2M−1[n] ∗rk[n]e − jnπ/M

(13)

and the upper-side ICI vanishes if the equalized channel im-pulse response is real in this baseband model Now the rela-tion between the decimated models is

z k[n]e − jn(k+1)π/M

n = mM =(−1)m(k+1) z k[mM] (14)

and a sufficient condition also for zero upper-side ICI is that

the equalized baseband channel impulse response is purely

real However, the baseband models for the two cases are

slightly different, and both conditions

Im



r k[n]

≡0,

Im



r k[n]e − jnπ/M

≡0

(15)

have to be simultaneously satisfied to achieve zero

over-all ICI In frequency domain, the equalized channel

fre-quency response is required to have symmetric amplitude

and antisymmetric phase with respect to both of the

fre-quencieskπ/M and (k + 1)π/M to suppress both ICI

com-ponents Naturally, the ideal full-band channel

equaliza-tion (resulting in constant amplitude and zero phase)

im-plies both conditions In our FBMC system, the

equal-ization is performed at low rate, after filtering and

dec-imation by M, and the mentioned two frequencies

cor-respond to 0 and π, that is, the filtered and

downsam-pled portion of Hch(e jω) in subchannel k multiplied by

the equalizer C k(e jω) must fulfill the symmetry condition

for zero ICI In this case, the two symmetry conditions

are equivalent (i.e., symmetric amplitude around 0 implies

symmetric amplitude around π, and antisymmetric phase

around 0 implies antisymmetric phase aroundπ) The

tar-get is to approximate ideal channel equalization over the

subchannel passband and transition bands with sufficient

accuracy

3.1.2 ISI analysis

In case of an ideal channel, the desired subchannel impulse

response of the baseband model can be written as



t k[n] =  t I k[n] + jt Q

k[n] = t k[n]e − jnkπ/M (16) For odd subchannels, a lowpass-to-highpass transformation

has to be included in the model to get the actual response for

the decimated filter bank, but the model above is suitable for

analyzing all subchannels Now the real part of the

subchan-nel response with actual chansubchan-nel and equalizer can be written

(see the appendix) as



g k[n] =Re



t k[n] ∗  r k[n]

=Re

t0[n] ∗  r k[n]

= t0I[n] ∗Re



r k[n]

− t Q0[n] ∗Im



r k[n]

.

(17)

The conditions for suppressing ICI are also sufficient for

sup-pressing the latter term of this equation Furthermore, in case

of PR filter bank design,t I

0[n] is a Nyquist pulse Designing

the channel equalizer to provide unit amplitude and

zero-phase response, a condition equivalent of having

Re



r k[n]

= δ[n] =

⎧

⎨

⎩

1, n =0,

0, otherwise, (18) would suppress the ISI within the subchannel

The above conditions were derived in the high-rate, full-band case, and if the conditions are fully satisfied, ISI within the subchannel and ICI from the lower and upper adja-cent subchannels are completely eliminated In practice, the equalization takes place at the decimated low sampling rate, and can be done only within the passband and transition band regions (assuming roll-off ρ =1.0) However, the ICI

and ISI components outside the equalization band are pro-portional to the stopband attenuation of the subchannel fil-ters and can be ignored

3.2 Optimization criteria for the equalizer coefficients

Our interest is in low-complexity subcarrier equalizers, which do not necessarily provide responses very close to the ideal in all cases Therefore, it is important to analyze the ICI and ISI effects with practical equalizers This can be carried out most conveniently in frequency domain In the baseband model, the lower and upper ICI spectrum magnitudes are



 V k Q(e jω)RQ

k(e jω)

=V Q

0(e jω)RQ

k(e jω)

= M

2



H p



e j(ω −(π/2M))

H p



e j(ω+(π/2M)) · R Q

k(e jω),



 U k Q

e jωR Q k



e j(ω+(π/M))

=U Q

2M−1

e jω R Q k

e j(ω+(π/M))

= M

2



H p



e j(ω −(π/2M))

H p

e j(ω+(π/2M))· R Q

k



e j(ω+(π/M)),

(19) respectively Here the upper-case symbols stand for the Fourier transforms of the impulse responses denoted by the corresponding lower-case symbols The terms involving the two frequency shifted prototype frequency responses are the overall magnitude response for the crosstalk.H p(e j(ω −(π/2M))) appears here as the receive filter for the desired subchan-nel andH p(e j(ω+(π/2M))) denotes the response of the trans-mit filter of the contiguous (potentially interfering) subchan-nel The actual frequency response includes phase terms, but based on the discussion in the previous subsection we know that, in the baseband model of the ideal channel case, all the cross-talk energy is in the imaginary part of the impulse response The residual imaginary part of the equalized channel impulse responserk Q[n] determines how

much of this cross-talk energy appears as ICI in detection

It can be calculated as a function of frequency for a given set of equalizer coefficients, assuming the required knowl-edge on the channel response is available Now the ICI power for subchannel k can be obtained with good

accu-racy by integrating over the transition bands in the baseband

P k ICI =

π/2M

− π/2M

M2

4



H p



e j(ω −(π/2M))

H p



e j(ω+(π/2M))2

· R Q

k

e jω2

dω

+

π/2M

− π/2M

M2

4



H p



e j(ω −(π/2M))

H p



e j(ω+(π/2M))2

· R Q k



e j(ω+(π/M))2

dω.

(20) Also the ISI power can be calculated, as soon as the

chan-nel and equalizer responses are known, from the aliased

spectrum ofGk(e jω), as

P ISI

π/M

0





M −

1

l =−1



G k

e j(ω+(lπ/M))





2

Here, the Nyquist criterion in frequency domain is used:

in ISI-free conditions, the folded spectrum of the overall

subchannel responseGk(e jω) adds up to a constant levelM, a

condition equivalent to overall impulse response being unity

impulse By calculating the difference between this ideal

ref-erence level and the actual spectrum, the spectrum resulting

from the residual ISI can be extracted Integration over this

residual spectrum gives the ISI power, according to (21)

Typically, the pulse shape applied to the symbol detector,

the slicer, is constrained to satisfy the Nyquist criterion In

the presence of ISI, this often requires from the receive filter

(in this context, the term “receive filter” is assumed to

in-clude both the analysis filter and the equalizer) a gain that

compensates for the channel loss and causes the noise power

to be amplified This is called noise enhancement The

sub-channel noise gain can be calculated as

β n

2π

π

− π



C k

e jω

H p



e j((ω ∓ π/2)/M)2

whereC k(e jω) is the response of the subchannel equalizer

The −and + signs are valid for even and odd subchannel

indexes, respectively

3.3 Semianalytical performance evaluation

The performance of the studied FBMC, using per-subcarrier

equalization to combat multipath distortion, can be

evalu-ated semianalytically according to the discussion above The

term “semianalytically” refers, in this context, to the fact that

no actual signal needs to be generated for transmission

In-stead, a frequency domain analysis of the distorting channel

and the equalizer can be applied to derive the ICI and ISI

power spectra and the noise enhancement involved Based

onP k ICI,P k ISI, andβ k n, the overall signal to interference plus

noise ratio(s) (SINR) for given E b /N0-value(s) can be

ob-tained Then, well-known formulas based on the Q-function

[30] and Gray-coding assumption can be exploited to

esti-mate the uncoded bit error-rate (BER) performance This

can further be averaged over a number of channel instances

corresponding to a given power delay profile

4 LOW-COMPLEXITY POINTWISE PER-SUBCARRIER EQUALIZATION

The known channel equalization solutions for FBMC suffer from insufficient performance, as in the case of the 0th-order ASCET [22], and/or from relatively high implementation complexity, as in the FIR filter based approach described, for example, by Hirosaki in [4] To overcome these problems, a specific structure that equalizes at certain frequency points

is considered The pointwise equalization principle proceeds from the consideration that the subchannel equalizers are designed to equalize the channel optimally at certain fre-quency points within the subband To be more precise, the coefficients of the equalizer are set such that, at all the con-sidered frequency points, the equalizer amplitude response optimally approaches the inverse of the determined chan-nel amplitude response and the equalizer phase response optimally approaches the negative of the determined chan-nel phase response Optimal equalization at all frequencies would implicitly fulfill the zero ICI conditions of (15), and the zero ISI condition of (18) In pointwise equalization, the optimal linear equalizer is approximated between the con-sidered points and the residual ICI and ISI interference pow-ers depend on the degree of inaccuracy with respect to the zero ICI/ISI conditions and can be measured using (20) and (21), respectively On the other hand, the level of inaccu-racy depends on the relation of the channel coherence band-width [31] to the size of the filter bank and the order of the pointwise per-subcarrier equalizer For mildly frequency selective subband responses, low-complexity structures are sufficient to keep the residual ICI and ISI at tolerable lev-els

Alternative optimization criteria are possible for the equalizer coefficients from the amplitude equalization point

of view, namely, zero-forcing (ZF) and mean-squared error (MSE) criteria [30,31] The most straightforward approach

is ZF, where the coefficients are set such that the achieved equalizer response compensates the channel response ex-actly at the predetermined frequency points The ZF crite-rion aims to minimize theP ICI k andP ISI k , but ignores the ef-fect of noise Ultimately, the goal is to minimize the proba-bility of decision errors The MSE criterion tries to achieve this goal by making a tradeoff between the noise enhance-ment and residual ISI at the slicer input The MSE criterion thus alleviates the noise enhancement problem of ZF and could provide improved performance for those subchannels that coincide with the deep notches in the channel frequency response For high SNR, the MSE solution of the ampli-tude equalizer converges to that obtained by the ZF crite-rion

4.1 Complex FIR equalizer

A straightforward way to perform equalization at certain fre-quency points within a subband is to use complex FIR fil-ter (CFIR-SCE), an example structure of which is shown in Figure 4, that has the desired frequency response at those given points In order to equalize for example at three

z 1 z 1

Re 

Figure 4: An example structure of the CFIR-SCE subcarrier

equal-izer

frequency points, a 3-tap complex FIR with noncausal

trans-fer function

HCFIR - SCE(z) = c −1z + c0+c1z −1 (23)

offers the needed degrees of freedom The equalizer

coef-ficients are calculated by evaluating the transfer function,

which is set to the desired response, at the chosen frequency

points and setting up an equation system that is solved for

the coefficients

4.2 Amplitude-phase equalizer

We consider a linear equalizer structure consisting of an

all-pass phase correction section and a linear-phase amplitude

equalizer section This structure is applied to each complex

subchannel signal for separately adjusting the amplitude and

phase This particular structure makes it possible to

indepen-dently design the amplitude equalization and phase

equaliza-tion parts, leading to simple algorithms for optimizing the

equalizer coefficients The orders of the equalizer stages are

chosen to obtain a low-complexity solution A few variants

of the filter structure have been studied and will be described

in the following

An example structure of the AP-SCE equalizer is

illus-trated in detail in Figure 5 In this case, each subchannel

equalizer comprises a cascade of a first-order complex

all-pass filter, a phase rotator combined with the operation of

taking the real part of the signal, and a first-order real allpass

filter for compensating the phase distortion The structure,

moreover, consists of a symmetric 5-tap FIR filter for

com-pensating the amplitude distortion Note that the operation

of taking the real part of the signal for detection is moved

before the real allpass phase correction stage This does not

affect the output of the AP-SCE, but reduces its

implementa-tion complexity

The transfer functions of the real and complex first-order

allpass filters are given by

H r(z) = 1 +b r z

H c(z) = 1− jb c z

1 +jb c z −1, (25) respectively In practice, these filters are realized in the causal

form as z −1H ·(z), but the above noncausal forms simplify

the following analysis For the considered example structure,

the overall phase response of the AP-SCE phase correction section (for thekth subchannel) can be derived from (24) and (25)

arg

Hpeq(e jω)

=arg

e jϕ0 · H c

e jω

· H r

e jω

= ϕ0 + 2 arctan



− b ckcosω

1 +b cksinω



+ 2 arctan



b rksinω

1 +b rkcosω



.

(26)

In a similar manner, we can express the transfer function of the amplitude equalizer section in a noncausal form as

Haeq(z) = a2z2+a1z + a0+a1z −1+a2z −2, (27) from which the equalizer magnitude response for the kth

subchannel is obtained



Haeq(e jω) =a

0 + 2a1 cosω + 2a2 cos 2ω. (28)

4.3 Low-complexity AP-SCE and CFIR-SCE

Case 1 The subchannel equalization is based on a single

fre-quency point located at the center frefre-quency of a specific subchannel, at ± π/2 at the low sampling rate Here the +

sign is valid for the even and the−sign is valid for the odd subchannel indexes, respectively In this case, the associated phase equalizer only has to comprise a complex coefficient

e jϕ0 for phase rotation The amplitude equalizer is reduced

to just one real coefficient as a scaling factor This case corre-sponds to the 0th-order ASCET or a single-tap CFIR-SCE

Case 2 Here, equalization at two frequency points located at

the edges of the passband of a specific subchannel, atω =0 andω = ± π, is expected to be sufficient The + and−signs are again valid for the even and odd subchannels, respec-tively In this case, the associated equalizer has to comprise, in addition to the complex coefficient e jϕ0, the first-order com-plex allpass filter as the phase equalizer, and a symmetric 3-tap FIR filter as the amplitude equalizer Compared to the equalizer structure ofFigure 5, the real allpass filter is omit-ted and the length of the 5-tap FIR filter is reduced to 3 In the CFIR-SCE approach, two taps are used

Case 3 Here, three frequency points are used for channel

equalization One frequency point is located at the center of the subchannel frequency band, atω = ± π/2, and two

fre-quency points are located at the passband edges of the sub-channel, atω =0 andω = ± π In this case, the associated

equalizer has to comprise all the components of the equalizer structure depicted inFigure 5 In the CFIR-SCE structure of Figure 4, all three taps are used

Mixed cases of phase and amplitude equalization Naturally,

also mixed cases of AP-SCE are possible, in which a different number of frequency points within a subband are considered for the compensation of phase and amplitude distortion For

b ck j

z 1

Complex allpass filter



b ck

j

z 1

e jϕ0

Re 

b rk 

z 1

z 1

Real allpass filter

z 1 z 1 z 1 z 1

a2 a1 a0 a1 a2

5-tap symmetric FIR

Phase equalizer Phase rotator

Amplitude equalizer

Figure 5: An example structure of the AP-SCE subcarrier equalizer

example,Case 3phase equalization could be combined with

Case 2amplitude correction and so forth Ideally, the

num-ber of frequency points considered within each subchannel is

not fixed in advance, but can be individually determined for

each subchannel based on the frequency domain channel

es-timates of each data block This enables the structure of each

subchannel equalizer to be controlled such that the

associ-ated subchannel response is equalized optimally at the

mini-mum number of frequency points which can be expected to

result in sufficient performance The CFIR-SCE cannot

pro-vide such mixed cases

Also further cases could be considered since additional

frequency points are expected to result in better performance

when the subband channel response is more selective

How-ever, this comes at the cost of increased complexity in

pro-cessing the data samples and much more complicated

for-mulas for obtaining the equalizer coefficients

ForCase 3 structure, CFIR-SCE and AP-SCE equalizer

coefficients can be calculated by evaluating (23) and (26),

and (28), respectively, at the frequency points of interest,

set-ting them equal to the target values, and solving the resulset-ting

system of equations for the equalizer coefficients:

CFIR-SCE:

c −1 = γ

4



χ0 − χ2

∓ j

2χ1 − χ0 − χ2



,

c0 = γ

2

χ0 +χ2

,

c1 = γ

4



χ0 − χ2

± j(2χ1 − χ0 − χ2 )

; (29)

AP-SCE:

ϕ0 = ξ0 +ξ2

b ck = ±tan



ξ2 − ξ0

4



,

b rk = ±tan



ξ1 − ϕ0

2



,

a0 = γ

4

0 + 21 +2

,

a1 = ± γ

4

0 − 2

,

a2 = γ

8

0 −21 +2

.

(30) Here the±signs are again for the even/odd

subchan-nels, respectively, andχ ,ξ , and  ,i = 0, , 2, are the

complex target response, the target phase, and amplitude re-sponse values at the three considered frequency points for subchannelk The value i =1 corresponds to the subchan-nel center frequency whereas valuesi =0 andi =2 refer to the lower and upper passband edge frequencies, respectively With MSE criterion,

χ ik = Hch

e j(2k+i)(π/2M) ∗



Hch

e j(2k+i)(π/2M)2

+η

,

ξ ik =arg

χ ik

,  ik =χ ik,

(31)

whereHchis the channel frequency response in the baseband model of the overall system The effect of noise enhance-ment is incorporated into the solution of the equalizer pa-rameters using the noise-to-signal ratioη and a scaling

fac-torγ =3/2

i =0χ ik Hch(e j(2k+i)(π/2M)) that normalizes the sub-channel signal power to avoid any scaling in the symbol val-ues used for detection In the case of ZF criterion,η =0 and

γ =1

The operation of the ZF-optimized amplitude and phase equalizer sections ofCase 3AP-SCE are illustrated with ran-domly selected subchannel responses in Figures6and7, re-spectively

InCase 2, MSE-optimized coefficients for CFIR-SCE and AP-SCE amplitude equalizer can be calculated as

c0 = γ

2

χ0 +χ2

,

c1 = ± γ

2

χ0 − χ2

,

a0 = γ

2

0 +2

,

a1 = ± γ

4

0 − 2

, (32)

whereγ =2/(χ0 Hch(e j(kπ/M)) +χ2 Hch(e j(2k+2)(π/2M))) The AP-SCE phase equalizer coefficients ϕ0 andb ck can be ob-tained as inCase 3

Case 1equalizers are obtained as special cases of the used structures, including only a single complex coefficient for CFIR-SCE and an amplitude scaling factor and a phase ro-tator for AP-SCE It is natural to calculate these coefficients based on the frequency response values at the subchannel center frequencies, that is,

c0 = χ1 ,

a0 =χ1 , ϕ0 =arg

χ1

withη =0, since MSE and ZF solutions are the same

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

Channel response

Equalizer target pointsε i

Equalizer amplitude response

Combined response of channel and equalizer

Normalized frequency (F s /2)

0

0.5

1

1.5

2

2.5

3

3.5

ε0

ε1

ε2

Figure 6: Operation of the ZF-optimizedCase 3amplitude

equal-izer section

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Channel response

Equalizer target pointsξ i

Equalizer phase response

Combined response of channel and equalizer

Normalized frequency (F s /2)

60

40

20

0

20

40

60

ξ2

Figure 7: Operation of theCase 3phase equalizer section

5 NUMERICAL RESULTS

The performance of the low-complexity subcarrier

equal-izers was evaluated with different number of subchannels

both semianalytically and using full simulations in time

do-main First, basic results are reported to illustrate how the

performance depends on the number of subcarriers and the

equalizer design case Also the reliability of the semianalytical

model is examined and the differences between ZF and MSE

criteria are compared Finally, more complete simulations

with error control coding are reported and compared to an OFDM reference in a realistic simulation environment Also sensitivity to timing and frequency offsets and performance with practical transmitter power amplifiers are investigated

We consider equally spaced real 2-PAM, 4-PAM, and 8-PAM constellations for FBMC and complex square-constellations QPSK, 16-QAM, and 64-QAM in the OFDM case

5.1 Semianalytical performance evaluation

Semianalytical simulations were carried out with the Vehicular-A power delay profile (PDP), defined by the rec-ommendations of the ITU [32], for a 20 MHz signal band-width These simulations were performed in quasi-static conditions, that is, the channel was time-invariant during each transmitted frame Perfect channel information was as-sumed In all the simulations, the average channel power gain was scaled to unity Performance was tested with filter banks consisting of 2M = {64, 128, 256}subchannels The filter bank designs used roll-off ρ = 1.0 and overlapping factor

K = 5 resulting in about 50 dB stopband attenuation The statistics are based on 2000 frame transmissions for each of which an independent channel realization was considered The semianalytical results were obtained by calculating the subcarrierwise ICI and ISI powersP ICI

k , respectively, together with noise gainsβ n

k fork =0, 1, , 2M −1 These were then used to determine the subcarrierwise SINR-values,

as a function of channelE b /N0-values, for all the channel in-stances The uncoded BER results were obtained for 2-, 4-, and 8-PAM modulations by evaluating first the theoretical subcarrierwise BERs based on the SINR-values using the Q-function and Gray-coding assumption, and finally averaging the BER over all the subchannels and 2000 channel instances

5.1.1 Basic results for AP-SCE

The comparison in Figure 8(a) for ZF 4-PAM shows that the time domain simulation-based (Sim) and semi-analytic model-based (SA) results match quite well This encourages

to carry out system performance evaluations, especially in the algorithm development phase, mostly using the semiana-lytical approach, which is computationally much faster Time domain simulation results inFigure 8(b)for 4-PAM indicate that the performance difference of ZF and MSE criteria is rather small Figures8(c)and8(d)show the semi-analytic re-sults for 2-PAM and 8-PAM, respectively, using the ZF crite-rion It can be observed that higher-order AP-SCE improves the equalizer performance significantly, allowing the use of a lower number of subcarriers Also ideal OFDM performance (without guard interval overhead) is shown as a reference With the aid of the AP-SCE equalizer, the performance of FBMC with a modest number of subcarriers can be made to approach that of the ideal OFDM

5.1.2 Comparison of CFIR-FBMC and AP-FBMC

In the other simulations, it is assumed that the receiver is time-synchronized such that the first path corresponds to

z... as-sumed In all the simulations, the average channel power gain was scaled to unity Performance was tested with filter banks consisting of 2M = {64, 128, 256}subchannels The filter bank